On radial behaviour and balanced Bloch functions

  • Juan J. Donaire

    Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain
  • Christian Pommerenke

    Technische Universität Berlin, Germany


A Bloch function gg is a function analytic in the unit disk such that (1z2)g(z)(1–|z|^2)|g' (z)| is bounded. First we generalize the theorem of Rohde that, for every "bad" Bloch function, g(rζ)(r1)g(r \zeta) (r \longrightarrow 1) follows any prescribed curve at a bounded distance for ζ\zeta in a set of Hausdorff dimension almost one. Then we introduce balanced Bloch functions. They are characterized by the fact that g(z)|g'(z)| does not vary much on each circle {z=r}\lbrace |z| = r\rbrace except for small exceptional arcs. We show e.g. that

01g(rζ)dr<\int^1_0|g'(r \zeta)|dr< \infty

holds either for all ζT\zeta \in \mathbb T or for none.

Cite this article

Juan J. Donaire, Christian Pommerenke, On radial behaviour and balanced Bloch functions. Rev. Mat. Iberoam. 15 (1999), no. 3, pp. 429–449

DOI 10.4171/RMI/261